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Properties

Label 2.1.2.2a1.1-1.4.12a
Base 2.1.2.2a1.1
Degree \(4\)
e \(4\)
f \(1\)
c \(12\)

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Defining polynomial

$x^{4} + b_{11} \pi^{3} x^{3} + \left(b_{10} \pi^{3} + b_{6} \pi^{2}\right) x^{2} + a_{9} \pi^{3} x + c_{12} \pi^{4} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: $\Q_{2}(\sqrt{-1})$
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $12$
Absolute Artin slopes: $[2,3,3]$
Swan slopes: $[3,3]$
Means: $\langle\frac{3}{2},\frac{9}{4}\rangle$
Rams: $(3,3)$
Field count: $8$ (complete)
Ambiguity: $2$
Mass: $8$
Absolute Mass: $4$

Diagrams

Varying

These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.

Galois group: $Q_8:C_2$ (show 4), $C_2^3 : C_4 $ (show 2), $C_2^4:C_6$ (show 2)
Hidden Artin slopes: $[\ ]^{2}$ (show 4), $[\ ]^{4}$ (show 2), $[2,2]^{3}$ (show 2)
Indices of inseparability: $[13,10,4,0]$ (show 2), $[13,12,4,0]$ (show 6)
Associated inertia: $[1,2]$ (show 6), $[1,3]$ (show 2)
Jump Set: $[1,2,13,21]$ (show 2), $[1,5,10,18]$ (show 2), $[1,5,15,23]$ (show 2), $[1,5,16,24]$ (show 2)

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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  displayed columns for results
Label Polynomial $/ \Q_p$ Galois group $/ \Q_p$ Galois degree $/ \Q_p$ $\#\Aut(K/\Q_p)$ Artin slope content $/ \Q_p$ Swan slope content $/ \Q_p$ Hidden Artin slopes $/ \Q_p$ Hidden Swan slopes $/ \Q_p$ Ind. of Insep. $/ \Q_p$ Assoc. Inertia $/ \Q_p$ Resid. Poly Jump Set
2.1.8.20d1.1 $x^{8} + 4 x^{5} + 2 x^{4} + 2$ $Q_8:C_2$ (as 8T11) $16$ $4$ $[2, 3, 3]^{2}$ $[1,2,2]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[13, 12, 4, 0]$ $[1, 2]$ $z^4 + 1,z^3 + 1$ $[1, 5, 10, 18]$
2.1.8.20d1.2 $x^{8} + 4 x^{5} + 2 x^{4} + 10$ $Q_8:C_2$ (as 8T11) $16$ $4$ $[2, 3, 3]^{2}$ $[1,2,2]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[13, 12, 4, 0]$ $[1, 2]$ $z^4 + 1,z^3 + 1$ $[1, 5, 16, 24]$
2.1.8.20d1.3 $x^{8} + 4 x^{7} + 4 x^{5} + 2 x^{4} + 2$ $C_2^3 : C_4 $ (as 8T19) $32$ $2$ $[2, 3, 3]^{4}$ $[1,2,2]^{4}$ $[\ ]^{4}$ $[\ ]^{4}$ $[13, 12, 4, 0]$ $[1, 2]$ $z^4 + 1,z^3 + 1$ $[1, 5, 15, 23]$
2.1.8.20d1.4 $x^{8} + 4 x^{6} + 4 x^{5} + 2 x^{4} + 2$ $C_2^3 : C_4 $ (as 8T19) $32$ $2$ $[2, 3, 3]^{4}$ $[1,2,2]^{4}$ $[\ ]^{4}$ $[\ ]^{4}$ $[13, 12, 4, 0]$ $[1, 2]$ $z^4 + 1,z^3 + 1$ $[1, 5, 15, 23]$
2.1.8.20d1.5 $x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 2 x^{4} + 2$ $Q_8:C_2$ (as 8T11) $16$ $4$ $[2, 3, 3]^{2}$ $[1,2,2]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[13, 12, 4, 0]$ $[1, 2]$ $z^4 + 1,z^3 + 1$ $[1, 5, 10, 18]$
2.1.8.20d1.6 $x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 2 x^{4} + 10$ $Q_8:C_2$ (as 8T11) $16$ $4$ $[2, 3, 3]^{2}$ $[1,2,2]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[13, 12, 4, 0]$ $[1, 2]$ $z^4 + 1,z^3 + 1$ $[1, 5, 16, 24]$
2.1.8.20d2.1 $x^{8} + 4 x^{5} + 2 x^{4} + 4 x^{2} + 2$ $C_2^4:C_6$ (as 8T33) $96$ $1$ $[2, 2, 2, 3, 3]^{3}$ $[1,1,1,2,2]^{3}$ $[2,2]^{3}$ $[1,1]^{3}$ $[13, 10, 4, 0]$ $[1, 3]$ $z^4 + 1,z^3 + z + 1$ $[1, 2, 13, 21]$
2.1.8.20d2.2 $x^{8} + 4 x^{7} + 4 x^{5} + 2 x^{4} + 4 x^{2} + 2$ $C_2^4:C_6$ (as 8T33) $96$ $1$ $[2, 2, 2, 3, 3]^{3}$ $[1,1,1,2,2]^{3}$ $[2,2]^{3}$ $[1,1]^{3}$ $[13, 10, 4, 0]$ $[1, 3]$ $z^4 + 1,z^3 + z + 1$ $[1, 2, 13, 21]$
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